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Updated: Jun 17, 2025

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Efficient Mixed-Precision Matrix Factorization of the Inverse Overlap Matrix in Electronic Structure Calculations

Adela Habib1, Joshua Finkelstein1, Anders M N Niklasson1

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|August 13, 2024
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Summary
This summary is machine-generated.

This study introduces a mixed precision algorithm for dense matrix factorization of the inverse overlap matrix (S^-1) using Nvidia Tensor cores. The method achieves high performance for electronic structure calculations, offering a robust approach for complex simulations.

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Area of Science:

  • Computational chemistry
  • Artificial intelligence hardware acceleration

Background:

  • Deep neural network calculations increasingly utilize AI hardware for high-performance tensor contractions.
  • Electronic structure theory requires efficient computation of the inverse overlap matrix (S^-1) for solving matrix eigenvalue problems.

Purpose of the Study:

  • To develop a mixed precision algorithm for dense matrix factorization of S^-1 using Nvidia Tensor cores.
  • To evaluate the performance and accuracy of this approach compared to traditional GPU implementations.

Main Methods:

  • Exploiting Nvidia Tensor cores for high-performance tensor contractions in reduced precision.
  • Developing a mixed precision iterative refinement algorithm for S^-1 factorization (ZZ^T = S^-1).
  • Implementing a nonparametric stopping criterion robust to lower precision floating-point operations.

Main Results:

  • Demonstrated high performance of Tensor cores for computing Z via matrix-matrix multiplications.
  • Compared the mixed precision approach against GPU-only single and double precision implementations.
  • Validated the robustness of the nonparametric stopping criterion in reduced precision.

Conclusions:

  • The mixed precision iterative refinement algorithm offers a high-performance solution for S^-1 factorization on Tensor cores.
  • This method is particularly beneficial for quantum-mechanical molecular dynamics and geometry optimization when good initial guesses are available.
  • The nonparametric stopping criterion ensures reliable computations despite potential precision limitations.